solow model, Macroeconomics

Q1. The poorest countries in the world have a per capita income of about $600 today.We can reasonably assume that it is nearly impossible to live on an income below halfthis level (i.e., below $300). Per capita income in Australia in 2010 was about $60,000.With this information in mind, consider the following questions.(a) For how long is it possible that per capita income in Australia has been growingat an average annual rate of 2% per year? (2 points)(b) Some economists have argued that growth rates are mismeasured. For example, itmay be difficult to compare per capita income today with per capita income acentury ago when so many of the goods we buy today were not available at anyprice. Suppose the true growth rate in the last two centuries was 3% per yearrather than 2%. What would the level of per capita income in 1850 have been inthis case? Is this answer plausible?

Q3. In this question, we are going to do some “normative” economics (i.e., “what oughtto be”) instead of “positive” economics (i.e., “what is”). Specifically, we will examinewhether the six countries in Q2 are investing too little or too much for the benefits oftheir future generations. For this question, again consider the Solow model with labourshare of 2/3rds.(a) Show mathematically that steady-state consumption per capita can be expressedas c* = A(k* )1/3 - dk* . Show your workings. (2 points)(b) Maximize steady-state consumption with respect to steady-state capital percapita—i.e., solve for ?c*?k* using the chain-rule in calculus that ?y?x= axa-1 for afunction y = xa . Denote the steady-state level of capital per capita that maximizessteady-state consumption per capita as kGR , where GR denotes “Golden Rule”(see below). What is kGR as a function of the productivity parameter and thedepreciation rate? (2 points)(c) Noting that steady-state capital will always be k* =sAd!" #$% &3/2for this model (why?),what s will maximize steady-state consumption (i.e., what value for s will makek* equal to the steady-state capital per capita that you solved in part (b))? (2points)(d) Macroeconomists refer to the value of s solved for in part (c) as the “GoldenRule” (i.e., “Do unto others,…”) investment rate. The idea is that investment atthis rate will maximize consumption for future generations. Meanwhile, a lowerinvestment rate means that households are consuming more today at the expenseof future generations, while a higher investment rate means that all generationsare investing too much and not enjoying consuming enough of the fruits of theirlabours. Based on the solution in part (c) and the investment rates in Q2, whichcountries are investing too little, too much, or just right, at least according to theSolow growth model and the Golden Rule investment rate? (2 points)